When did we first fall in love with mathematics? For me, it was in class 6.
My father exposed me to a problem from Euclidean geometry. We were traveling in Kausani.
After days of frustration and failed attempts, I could put together the ‘reason’ that made ‘everything fit together perfectly’. The problem was solved and beauty of ‘pure reason’ revealed itself. It was breathtaking. I fell in love!
This has been the guiding principle in my teaching efforts. At the core of mathematics is ‘reason’. We definitely draw inspiration from real world observations. However, one does mathematics because he or she adores ‘reason’ itselfand not the observations.
Years later, I was exposed to Kant’s ‘Critique of Pure Reason’ (thanks to my wife). I was tempted to switch to Philosophy. The promise to go beyond reason was alluring.
Methodology
The philosophical foundation of this eight - week course in beautiful mathematics is therefore well-grounded in these personal experiences. I hope to expose the students to the enchanting beauty of ‘reason’. This is planned in the following manner:
Begin with some observations of objects
After repeated observations, find a pattern. A pattern is, roughly speaking, Plato’s ‘form’. It is something that reveals itself when you threw away all the ‘unimportant elements’ from your observations.
Once observation and pattern recognition are accomplished, one employs ‘reasoning’ to see why the pattern could be relevant in broader contexts.
Recognize pattern by counting vertices, faces, and edges (Euler’s number)
Generalize that Euler number is, in essence, an ‘invariance phenomena’. Invariance is omnipresent. Employ that to understand golden ratio (that is see the application of invariance principle in a completely different context)
There are two other things, that I would love to try in this course
Dialectical investigation in the line of Tarasov. Roughly speaking, this appeals to two steps
deconstruction of a big idea into fundamental pieces
reconstruction of the big idea from those pieces.
Rabindranath’s experiments with pedagogy, especially relating to the objects of observation. Rabindranath recognized that if one stays close to nature and social fabric at the observation stage, then the pattern recognition and generalizations are fundamentally altered. Though I have not experimented or studied this claim in detail, it seems plausible.
Cheenta - Filix Level 1 Math Olympiad Starter module.
Day 0 - Warm up with beautiful problems and drawings.
Day 1 - Platonic Solids (Cube, Tetrahedron, Octahedron, projections)
Day 2 - Platonic Solids (Icosahedron, Dodecahedron, projections)
Day 3 - Counting the simplexes
Day 4 - Invariance principle (Euler number)
Day 5 - Invariance principle (Golden ratio)
Day 6 - Algorithms (Fibonacci number generator)
Day 7 - Algorithms (Fibonacci number generator)
Day 8 - General problems from invariance principle
How the sessions are designed?
Each session begins with a ‘Motivation problem sheet’. Students are expected to try these problems on their, possibly even before attending the class. They are allowed and encouraged to discuss amongst themselves.
The discussion kickstarts with a big problem or big idea. Lectures are limited to 15 minute slots. Students will need to ‘do’ mathematics after each such 15 minute slot.
The session ends with a ‘Follow up problem sheet’. These problems are ‘collaborative homework’. They are most effective when students discuss them in groups.
The faculty may recommend some additional reading!
